Two-part ratios feel comfortable. You see "2 : 3" and you know what to do — two parts for one, three for the other, five parts total. But then a problem shows up with three terms: "share ₹6000 in the ratio 3 : 4 : 5 between Asha, Bhavna, and Chetan." Suddenly the familiar picture has a third piece, and you wonder whether the rules still apply the same way.

They do. A three-term ratio is exactly a two-term ratio with one more participant, and the splitting procedure generalises without any change. You just have to trust that the sum of the terms is still where the "total parts" live.

What 3 : 4 : 5 actually says

The ratio 3 : 4 : 5 says: "for every 3 units the first person gets, the second gets 4, and the third gets 5." So if the first person gets 3 mangoes, the second gets 4, and the third gets 5 — together they share 12 mangoes.

If they share 24 mangoes, the first gets 6, the second 8, the third 10. Still 3 : 4 : 5. The shape is fixed; the size depends on the total.

The key fact — the one that makes every three-part ratio solvable — is that the total divides into 3 + 4 + 5 = 12 equal "parts." Each person's share is the number of parts in their ratio term, times the value of one part.

Six thousand rupees split in the ratio three to four to five among Asha, Bhavna, and ChetanA horizontal bar representing six thousand rupees divided into twelve equal parts of five hundred rupees each. The first three parts are shaded in one colour and labelled Asha. The next four parts are shaded in another colour and labelled Bhavna. The last five parts are shaded in a third colour and labelled Chetan. Totals beneath: Asha one thousand five hundred, Bhavna two thousand, Chetan two thousand five hundred. 3 parts — Asha 4 parts — Bhavna 5 parts — Chetan ₹1500 ₹2000 ₹2500 12 parts × ₹500 each = ₹6000 each tick = one part = ₹500
Six thousand rupees divided into twelve equal parts of ₹$500$ each — three parts for Asha, four for Bhavna, and five for Chetan, matching the ratio $3 : 4 : 5$. The tick marks make the "one part" unit visible: once you know one part is worth ₹$500$, each share is a simple multiplication.

The three-step method

The procedure is the unitary method applied to a bigger stage. It has exactly three moves.

Step 1. Add all the ratio terms to find the total number of parts.

For 3 : 4 : 5, the sum is 3 + 4 + 5 = 12.

Why: the ratio says the total is divided into groups of 3, 4, and 5 units respectively. These groups fit end-to-end without overlap, so the total number of units is the sum of the three groups.

Step 2. Divide the total quantity by the number of parts to find the value of one part.

\text{one part} = \frac{6000}{12} = 500

So one "part" is worth ₹500 in this problem. This is the unitary step — the single bridge that converts a proportion into actual numbers.

Step 3. Multiply one part by each ratio term to find each person's share.

\text{Asha} = 3 \times 500 = 1500
\text{Bhavna} = 4 \times 500 = 2000
\text{Chetan} = 5 \times 500 = 2500

And a built-in sanity check: the three shares must add back to the original total.

1500 + 2000 + 2500 = 6000 \,\checkmark

If they don't, you made an arithmetic slip somewhere and should retrace.

As fractions instead of parts

There is a second way to write the same procedure, using fractions of the total. Each person's share is their ratio term divided by the sum of the ratio terms, times the total.

\text{Asha} = \frac{3}{12} \times 6000 = 1500
\text{Bhavna} = \frac{4}{12} \times 6000 = 2000
\text{Chetan} = \frac{5}{12} \times 6000 = 2500

This is algebraically identical to the three-step method — it just combines the "divide by 12" and "multiply by the term" into one fraction. Some people find fractions faster; some find parts faster. Use whichever you spot first.

Why the fractions: the denominator 12 is the total number of parts, and the numerator is how many of those parts belong to that person. So \tfrac{3}{12} is the fraction of the whole that Asha gets, \tfrac{4}{12} is Bhavna's share, and \tfrac{5}{12} is Chetan's.

A trickier example: when the total is unknown

Sometimes the problem gives you one person's share instead of the total, and asks you to fill in the rest.

"Asha, Bhavna, and Chetan split a bonus in the ratio 3 : 4 : 5. Asha received ₹1800. How much did each of the others receive, and what was the total bonus?"

You cannot use the three-step method directly because you do not know the total. But you can find the value of one part from Asha's share: Asha's 3 parts are worth ₹1800, so one part is 1800 / 3 = ₹600. Now multiply: Bhavna gets 4 \times 600 = 2400, Chetan gets 5 \times 600 = 3000, and the total bonus is 3 \times 600 + 4 \times 600 + 5 \times 600 = 12 \times 600 = 7200.

The lesson: any piece of information that pins down one part is enough. You do not need the grand total specifically. You need some way to figure out what one part is worth, and then everything else cascades.

Common mistakes to avoid

Mistake 1. Dividing by the wrong denominator. The most common error is dividing the total by the first ratio term, or by one of the terms, instead of by the sum. If you wrote 6000 / 3 = 2000 and called that Asha's share, you have used the wrong denominator. The sum 3 + 4 + 5 = 12 is the correct divisor for finding the unit part, and any other number will give wrong shares.

Mistake 2. Forgetting to include all ratio terms in the sum. With a 3 : 4 ratio, the sum is 7. With a 3 : 4 : 5 ratio, the sum is 12. This sounds obvious, but under exam pressure students sometimes use 7 (the sum of the first two) instead of 12 (the full sum). All ratio terms contribute to "parts-of-the-whole."

Mistake 3. Treating "the ratio 3 : 4 : 5" as "the first has 3, the second has 4, the third has 5." No — those are the relative sizes. The actual amounts depend on the total, as above.

The same recipe for any number of terms

The procedure generalises unchanged to four, five, or more terms. Share ₹10{,}000 in the ratio 1 : 2 : 3 : 4? Add the terms: 1 + 2 + 3 + 4 = 10 parts. One part is 10000 / 10 = 1000. Each person's share is 1000, 2000, 3000, 4000. That is the entire technique.

Once you internalise this, ratio questions become a two-line reflex: sum the terms; divide the total by the sum; multiply. The number of terms never changes the method — only the arithmetic.

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